Curved Rectangular Beam Stress Formulas and Calculator

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Curved Rectangular Beam Stress Formulas and Calculator

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Curved Rectangular Beam Stress Formulas and Calculator

Bending stresses in curved beams do not follow the same linear variation as straight beams, because of the variation in arc length. Even though the same assumptions are used for both types, i.e. that plane section perpendicular to the axis of the beam remain plane after bending and that stress is quite different. Figure 1.0 shows the linear stress variation in a straight beam and the hyperbolic stress distribution in a curved beam. Note that the bending stress in the curved beam is zero at a point other than at the center of gravity axis. Also, note that the neutral axis is located between the gravity axis and the center of curvature: this always occurs in curved beams.

Figure 1.0

Stress Distribution due to bending is given by:

Equation 1
s = M · γ / ( A · e · ( r_n - y ) )

Where:

s = the bending stress, psi
M = bending moment with respect to the centroical axis, in-lb
y = distance from the neutral axis to the point in question, inches (positive for distances toward the center of curvature, negative for distances away from the center of curvature)
A = the area of the section, in²
e = distance from the center of gravity axis to the neutral axis, inches
r_n = radius of curvature of the neutral axis, inches

Bending Stress at the Inside Fiber is given by:

Equation 2
s = ( M · h_i ) / ( A · e · r_i )

Where:

h_i = distance from the center neutral axis to the inside fiber, inches ( h_i = r_n - r_i )
r_i = radius of curvature on the inside fiber, inches

Bending Stress at the Outside Fiber is given by:

Equation 3
s = ( M · h_o ) / ( A · e · r_i )

Where:

h_o = distance from the center neutral axis to the inside fiber, inches ( h_o = r_o - r_i )
r_o = radius of curvature on the outside fiber, inches

Figure 2.0 Rectangular Beam

Rectangular shape

r_n = h / ( log_e · r_o / r_i )

log_e = 0.434294481

e = R - r_n

R = r_i + h / 2

References:

McGraw Hill Machine Design (1968)